The Behavior of Structures Composed of Composite Materials Part 9 ppt

231 The terms are omitted in the above for brevity, and Stresses in each lamina are calculated at two axial locations to show the stress distributions that occur in the bending boundary layer. Stresses are calculated at an axial location where the deflection is on half of the imposed edge displacement, i.e. Secondly, stresses are calculated at an axial location of x = l/2, where l is the axial distance at which w = 0, nearest to the end of the shell. Results are shown in Figure 5.6. It is seen that the stresses are tensile on the outer layers and compressive on the inner layers. It is also seen that the circumferential stresses, are maximum in the layers, which is logical because the circumferential stiffness is greatest in these layers and hence the circumferential load is carried in the 90° layers analogous to a stiff spring in a collection of parallel springs. Lastly, even though the deflections and stresses are axially symmetric, in-plane shear stresses, do exist because of the stacking sequence of the laminate. 232 5.6 Sample Solutions 5.6.1 EFFECTS OF SIMPLE AND CLAMPED SUPPORTS Consider the circular cylindrical shell composed of a transversely isotropic composite shown in Figure 5.7, typical of many RTM fabric composites today. The end of the shell x = 0 is simply supported, the end x = L is clamped. The ends of the shell are assumed to be rigid plates. The internal pressure is and is written simply as v in this example. The following information is desired: 233 What is the magnitude and exact location of the maximum stress that occurs in the bending boundary layer at the simply supported end? The same information near the clamped end? What is the magnitude of the maximum stress occurring outside the bending boundary layers, say at x = L /2? If the maximum principal stress failure theory is used, would the shell be structurally sound if in the design the thickness had been determined by membrane shell theory? (Membrane shell theory neglects all bending effects, see Section 5.10). a. b. c. d. From the external axial force equilibrium, The boundary conditions at x = 0 are w (0) = 0 and M (0) = 0. From Equations (5.56) and (5.59), the boundary conditions are: Hence, where because of the composite configuration, the material is quasi-isotropic and for this example for simplicity. It is worthwhile to point out that for a cylindrical shell closed at the ends by a structure of any shape, when there is an internal pressure, then and this appears in the (– v /2) terms in equations (5.63), (5.65), (5.73) and (5.74). If the cylindrical shell were a nozzle (i.e. no ends), then and the (– v / 2) term would not appear. 234 To determine the location of the maximum value of the location of the maximum value of M (x) is required, since is a constant. Since is zero at x = 0, and again it is zero for extreme values lie somewhere in the bending boundary layer. They occur at values of x where Since therefore the extrema occur at From Equation (5.64) and (5.65), the fact that and since the shell is long, the effects of and are negligible. Equation (5.60) becomes For this condition to exist, or etc . Since there is an exponential decay with increasing x, the largest extreme value, the maximum, occurs at Hence, Equations (5.35) and (5.36) can be used instead of (5.62) since the shell is made of a transversely isotopic composite material, such as many RTM fabric composites. From Equations (5.64), (5.65), (5.56) and (5.59) Therefore from Equations (5.18), (5.36), (5.62), (5.68) and (5.69) 235 For v = 0.3, this reduces to Extreme values occur for the condition which requires that The + requirement occurs for the negative when Of these two, is a maximum in the former case. The maximum value of in the range is At the clamped end, the boundary conditions are From Equations (5.56) and (5.58) and making use of the fact that the shell is long Hence, 236 It can be shown and is physically obvious that at the clamped end, Hence, at x = L , However, occurs away from the end of the shell. Analogous to the procedures used at the previous end, it is found that At x = L / 2, which is outside either bending boundary layer, it is seen from Equation (5.35) that hence, which is recognized as the membrane solution. Likewise, This is also the membrane solution. It is seen that the maximum principal stress occurring in the boundary layer at the simply supported end is, from Equation (5.72) correspondingly, at the clamped end it is [see (5.75)]. The maximum stress predicted by membrane theory is in the hoop direction seen in (5.78). Hence, stresses greater than the membrane stresses occur in both boundary layers; 10% higher in the simply supported area, and 104% higher in the clamped edge. Thus, this shell if designed on a basis of membrane shell theory would be woefully inadequate. In the above example, the location of the maximum axial stress and the maximum circumferential stress at each end of the shell have been determined. It should be remembered that a biaxial stress state exists everywhere. Hence, when dealing with a material which follows yield or fracture criteria such as a maximum distortion energy criteria, maximum shear stress, or one of several others, then that criteria must be used to find the location of the maximum value of the equivalent uniaxial stress state as discussed in Chapter 7. Quite often a very simple digital 237 computer routine can be employed to do the arithmetic, using the analytical solution, to determine the location and magnitude of the maximum stress state. As pictorial examples of the stress couples, are the transverse shear resultants, that exist in the bending boundary layers, Figures 5.8 and 5.9 illustrate that for a simply supported edge at x = 0 the stress couple is zero at the edge and peaks a short distance away, while the maximum value of the shear resultant occurs at the edge. Both and go to zero in the bending boundary layer, and at greater distances from the edge only membrane stresses and deflections occur. Figures 5.10 and 5.11 show analogous results for the clamped edge at x = L = 200 inches where it is seen that both the stress couple, and the transverse shear resultant, are maximum at the clamped edge, but diminish to zero in the bending boundary layer. Figures 5.8 through 5.11 are from a paper by Preissner [3] in which the values of aare defined by (5.88) in the next section, and was defined also in Chapter 4 for an asymmetric composite beam. 238 239 5.7 Mid-Plane Asymmetric Circular Cylindrical Shells In many applications there are good reasons to have the shell structure be mid-plane asymmetric with respect to the materials used. The exterior environment may differ markedly from the interior environment to which the shell is exposed the outer environment may have extremes of temperature and humidity, blast damage, etc. while the inner environment may have chemical, abrasive, esthetic or other considerations. Also stealth considerations may play a role in some shell structures. 5.7.1 GOVERNING DIFFERENTIAL EQUATIONS To study the shell with mid-plane asymmetry under axially symmetric loads, the equilibrium equations and the strain displacement relations remain the same as in Section 5.2.2, in Equation (5.19) through (5.28). However, the constitutive equations for the mid-plane asymmetric composite, which is specially orthotropic so that the material axes 1-2-3 coincide with the axes are found from Equation (2.66) to be 240 As before, substituting derivatives of these results into the equilibrium equations and using the strain-displacement relations results in the following equation analogous to Equation (5.34). The second governing differential equation analogous to Equation (5.33) is Defining a reduced or effective flexural stiffness as: and introducing the following definition Equation (5.84) can now be written as: Except for the second term on the left-hand side of Equation (5.87), this is the usual governing equation for an axially symmetric cylindrical shell of flexural stiffness subjected to axially symmetric loads as given in Equation (5.33). It is also noted that the second term of Equation (5.87) varies directly with the asymmetry quantities, and Recall that a mid-plane asymmetry parameter was defined previously in Chapter 4 as [...]... Vibration of Laminated Orthotropic Cylindrical Shells, Journal of the Acoustical Society of America, Vol 44, pp 1628-1635 8 Bert, C.W., Baker, J.L and Egle, D ( 196 9) Free Vibrations of Multilayer Anisotropic Cylindrical Shells, Journal of Composite Materials, Vol 3, pp 480- 499 9 Tasi, J ( 196 8) Reflection of Extensional Waves at the End of a Thin Cylindrical Shell, Journal of the Acoustical Society of America,... pp 1 791 -1 797 18 Whitney, J.M and Sun, C.T ( 197 5) Buckling of Composite Cylindrical Characterization Specimen, Journal of Composite Materials, Vol 9, April, pp 138-148 19 Hsu, Y.S., Reddy, J.N and Bert, C.W ( 198 1) Thermoelasticity of Circular Cylindrical Shells Laminated of Bimodulus Composite Materials, Journal of Thermal Stresses, Vol 4, pp 155-177 20 Simitses, G.J., Shaw, D and Sheinman, I ( 198 5)... elastic body component of the surface traction component of the deformation component of a body force of the body surface over which tractions are prescribed We see that the first term on the right-hand side of Equation (6.1) is the strain energy of the elastic body The second and third terms are the work done by the surface tractions; and the body forces, respectively The Theorem of Minimum Potential Energy... cylindrical shell of length radius and a wall thickness composed of a unidirectional composite of properties and What is the length of the bending boundary layer? 5.3 For a circular cylindrical shell, composed of a the unidirectional graphite-epoxy given in Problem 5.1 for a shell of radius wall thickness of and length what is the length of the bending boundary layer if: (a) The fibers are in the axial direction?... cylindrical shell of dimensions and composed of 4 laminae of T300 /93 4 graphite/epoxy, each thick, with the fibers placed in the axial direction What is the critical value of Torque, the shell can withstand without buckling? Is the restriction on the equation satisfied? 5 .9 For a long composite shell of a unidirectional composite material subjected to a uniform external pressure, to attain the highest buckling... M.A ( 196 2) New Techniques of Solution for Problems in the theory of Orthotropic Plates, Transactions of the Fourth U.S National Congress of Applied Mechanics, Vol 2, pp 817-825 5 Anon ( 196 8) Buckling of Thin Walled Circular Cylinders, NASA SP-8007, Revised, August 6 White, J.C ( 196 1) The Flexural Vibrations of Thin Laminated Cylinders, Journal of Engineering Industry, pp 397 -402 7 Dong, S.B ( 196 8) Free... length composed of the same material as in Problem 5.1 wherein and What is the length of the bending boundary layer for this shell? 5.6 Consider a shell composed of a quasi-isotropic composite material wherein psi and v = 0.3 The mean shell radius is 10 inches and the total thickness is h = 0.3 inches The length is 40 inches, and the interlaminar shear strength sufficient (a) What is the length of the. .. 44, pp 291 - 292 10 Tasi, J ( 197 1) Effect of Heterogeneity on the Axisymmetric Vibration of Cylindrical Shells, Journal of Sound and Vibration, Vol 14, pp 325-328 11 Tasi, J and Roy, B.N ( 197 1) Axisymmetric Vibration of Finite Heterogeneous Cylindrical Shells, Journal of Sound and Vibration, Vol 17, pp 83 -94 12 Mirsky, I ( 196 4) Vibration of Orthotropic, Thick, Cylindrical Shells, Journal of the Acoustical... (a) What is the overall buckling load (b) How many lbs of compressive load can the shell withstand before buckling? (c) What are the values of the integers m and n corresponding to the minimum buckling load? (d) What is the value of the axial stress? ? (e) What is the membrane hoop stress As the compressive load is applied, will the shell buckle first or be overstressed? (f) 5.11 A piece of composite. .. (5.123) The procedure to follow is equivalent to that of Sections 5.8.6 and 5.8.7, except in checking for overstressing The non-zero loads to employ with Equations (5.1 09) and (5.110) are: 5 .9 Vibration of Composite Shells The natural and forced vibration of composite shell structures is very involved, time consuming to present and study, and therefore is left to the reader Given here are some of the basic . for the condition which requires that The + requirement occurs for the negative when Of these two, is a maximum in the former case. The maximum value of in the range is At the clamped end, the. were a nozzle (i.e. no ends), then and the (– v / 2) term would not appear. 234 To determine the location of the maximum value of the location of the maximum value of M (x) is required, since. canceling the terms it is found for n = 0: and for 243 The forcing function of the n = 0 equation, (5 .95 ), is the actual applied load to the shell. For the forcing function involves derivatives of the